The Modular Group and Hyperbolic Space

That is, the modular group is the set of Möbius transformations that have integer coefficients and coefficient matrices with determinants equal to one.

What is a group? Recall that a group is a set G together with a binary operation satisfying certain properties: the set must be closed under the operation and the operation must be associative. There must also be an identity element for the operation, and all inverses of elements in G must themselves be elements of G. The proof that the modular group is, in fact, a group is a standard exercise in a course on complex analysis [2, 277-8].

We will assume that the elements of the modular group do indeed form a group and investigate some of its interesting subgroups.

Fundamental Regions

One of our main goals is to investigate how the elements of the modular group act on fundamental regions, that is, how the regions are stretched and bent when we view them as Euclidean objects. Indeed, as hyperbolic objects, the regions are all carbon copies of each other, in much the same way that the squares on a checkerboard are all identical.

In general, a group of one-to-one transformations acting on a topological space partitions that space into fundamental regions. For a collection of sets {} to be a collection of fundamental regions, certain properties must hold. First and foremost, the must be pairwise disjoint. Second, given any transformation f in the group, , unless f is the identity map. Finally, given any two regions and , there exists some transformation f, such that .

Generally, in order to cover the entire space disjointly, each fundamental region must contain some but not all of its boundary points. This is a technical matter with which we will not concern ourselves.

In fact, we will relax the above definition so that we may include all of the boundary points when we consider any particular fundamental region. Thus, we will allow the intersection of adjacent fundamental regions to overlap, but only on their boundaries. The essential feature remains that there is no area in the overlap of adjacent regions.

Note also that a group of transformations does not necessarily yield a unique partition of the space into fundamental regions. Thus, the following fundamental regions are merely two representative fundamental regions.

Figure 4. Two fundamental regions and a single region with vertices marked.

Each fundamental region is made up of a shaded region with an unshaded region outlined with the same color. One of the regions is enlarged on the right to identify vertices and indicate a method to measure angles. Each fundamental region contains four vertices that can be fixed (or left in place) by elements of the modular group. The illustration on the right has its four vertices identified with red dots. Tangents are drawn at one vertex to indicate how the angle measures can be determined. The left and right vertices of the region have 60-degree angles. The vertex at the top has a straight, 180-degree angle. The vertex at the bottom has a zero-degree angle because the tangents to the intersecting arcs coincide at that point. Any hyperbolic polygon with a vertex on the boundary of the space, the x-axis in this case, will have a zero angle at that vertex. The corresponding four angles in each fundamental region have the same measures as those indicated here. Each vertex can be fixed by some element in the modular group. Further, each fundamental region can be mapped onto any other fundamental region by an element of the modular group.

A classic view of the matter is to see the upper half-plane as tessellated or tiled by triangular-shaped regions, as in the first illustration of this article. A checkerboard tessellation of the Euclidean plane can be constructed by sliding copies of a square to the left, right, up, and down. Eventually the plane is covered with square tiles. The modular group tessellates the hyperbolic plane in an analogous way. The elements of the group move copies of a fundamental region until triangular-shaped tiles cover the upper half-plane model of the hyperbolic plane. Of course these tiles do not appear to be identical to our eyes that are trained to match shapes and lengths in Euclidean geometry. However, the triangular-shaped tiles are all identical if measured using the hyperbolic metric. In the tiling process, all areas in the upper half-plane are covered by tiles and no tiles have any overlapping areas. In fact, this procedure is precisely how the hyperbolic plane illustration was constructed. The boundary points for a single fundamental region were acted on by function elements of the modular group and the resulting points were drawn as a polygon or boundary line in the illustration.

It helps to note that each transformation in M has at least one fixed point. Some transformations in M have two fixed points. Only the identity map has more than two. In the illustrations that follow, we will observe the placement of fixed points and the way transformations map fundamental regions near those fixed points.

Hyperbolic Lengths

The hyperbolic metric is a rather curious metric that challenges our notion of "distance." Under the hyperbolic metric, the shortest distance between two points is seldom along a straight line, but rather along an arc of a circle. For example, in the upper half-plane, the shortest (hyperbolic) path between points and is the top arc of the circle that passes through both points and is perpendicular to the real axis.

Figure 5. The shortest hyperbolic path between the points and .

Without discussing precisely how hyperbolic lengths and areas are measured, we state that every image under a transformation in the modular group is congruent to every other image under the hyperbolic metric. Thus, all of our fundamental regions shown in the animations are actually the same "size" in the hyperbolic metric.